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In this section, I show that starting from the least-squares
inverse of the subtraction method, I can retrieve a fitting goal that
resembles the fitting goal of the filtering technique.
I first assume that the data is the sum of
the signal and the noise as follows:
| |
(1) |

Now I assume that I have two modeling operators
and for the signal and noise respectively such
that
| |
(2) |

where and are unknown model parameters.
In this paper I assume that the two modeling operators are
orthogonal, meaning that they model different regions of the data space.
Inserting the modeling operators into equation (1) I have
| |
(3) |

I then want to estimate and so that
| |
(4) |

In a least-squares sense, I have to minimize the following objective
function:
| |
(5) |

The least-squares inverse of the model parameters is given by Guitton (2001)
with
| |
(6) |

One can easily recognize in the definition of the identity matrix minus the signal resolution operator and for , the identity matrix minus the noise resolution operator.
Therefore, and are
signal and noise filtering operators respectively with
| |
(7) |

An interesting property of and
is that and : they
are called projectors. It is easy to verify that they are
Hermitian operators as well Guitton (2001).
Now, looking at the first row in equation(6), I have
| |
(8) |

This expression is the least-squares inverse for the following
objective function
| |
(9) |

which can be rewritten
| |
(10) |

or
| |
(11) |

using the properties of . Therefore,
equation (12) is the objective function for the fitting goal
| |
(12) |

Similarly, if I look at the second row of equation (6), I
end up with
| |
(13) |

** Next:** Why is it working
** Up:** Guitton: Signal/noise separation
** Previous:** Introduction
Stanford Exploration Project

7/8/2003